Tywin defines "legacy" as "It's what remains of you when you're gone." Then he tells her about Harrenhal, which had the tallest towers and strongest walls. Now it's a blasted ruin. What happened? "Dragons happened." The problem was that Harrenhal was built to withstand an attack from the land. Tywin says, "A million men could have marched on these walls and a million men would have been repelled." Instead, Aegon Targaryen changed the rules (attacked with dragons), so everyone knows his name 300 years later. Arya points out that Argon's sisters Rhaenys and Visenya were also riding dragons. I like that she's more interested in the women. She provides biographical information: Rhaenys rode Meraxes. Visenya rode Vhagar. Tywin: "I'm sure I knew that when I was a boy." I also like that she can't remember noble houses and sigils, but she's good at who rode what dragon. Arya talks about how great Visenya was. Tywin asks if girls aren't supposed to be interested in pretty maidens from songs. Arya utters what might as well be her motto: "Most girls are idiots." He asks where she learned about Visenya and her Valyrian steel sword. She again claims it was her father. Tywin observes that he's never met a literate stonemason. She shoots back, "Have you met many stonemasons, my lord?" He glares at her and says, "I enjoy you, but be careful." He has her take the food back to the kitchen and eat it there. And he tells her that lowborn girls say "M'Lord," not "My lord." Arya claims that her mother served Lady Dustin for many years. And as a result: "She taught me how to speak proper -- properly." Tywin tells her she's too smart for her own good. She admits that she's been told that before.
It seems to me that Tywin knows she's lying about where she's from. He doesn't know she's Arya Stark, though. And he's enjoying the fact that he's got someone smart to talk to, but he also doesn't like it when someone gets off a good one at his expense.
Sansa passes the Hound in the halls of King's Landing. She thanks him for saving her from those would-be rapists and calls him brave. He says it's not brave for a dog to scare off rats. He says he enjoys killing, and that it's the sweetest thing there is. Even Ned Stark liked killing people, he says. Sansa doesn't like the sound of that. The Hound says she'll be thankful for him someday, when she's queen and he's all that stands between her and Joffrey.